VISCOSITYVISCOSITY

VISCOSITY OF WATER BY CAPILLARY FLOW

REFERENCES

D.M. Burns and S.G.G. MacDonald, Physics 2nd edition, (Copies available at the Resource Centre)J.W. Kane and M.M. Sternheim, General Physics.D. Halliday, R. Resnick and J. Walker, Physics.

If you have not yet encountered the concept of viscosity in your studies, do read therelevant introduction to the subject in one of the references.

INTRODUCTION

The coefficient of viscosity of a fluidcan be found from measurements ofthe volume rate of flow through smalltubes. For simple analysis to bepossible laminar or streamlined flowmust prevail. Conditions of flow canbe described by a dimensionlessquantity, Reynolds Number, NR.NR is defined as:

NR��vD�

where � is the density, v an averagevelocity of flow, D a linear dimensionfor the system in question, and � thecoefficient of viscosity. In the case offluid flow in a tube D is the diameterof the tube, and for laminar flow NR

must be less than about 2000, i.e., thevelocity of flow in the tube must belimited. At higher velocities the flowbecomes turbulent.

For streamlined flow of a liquid through a tube the volume rate of flow, Q, is given by Poiseuille'sformula:

, or, transposing, Q ��a 4

8��p p �

8��

�a 4Q

where a is the radius of the tube, � its length, and p the pressure difference across its length. Notethat this formula reflects the physical equilibrium situation of a force on the fluid due to a pressuredifference p being balanced by a viscous force (i.e., one due to a frictional effect), with no otherforces acting. (J.L.M. Poiseuille, a French physician interested in the circulation of the blood,experimentally determined the formula in 1835.)

VISCOSITY

EXPERIMENT

The flow of water, Q, is to be measured atseveral values of p and for two or threedifferent capillary tubes (available at theResource Centre). The pressure p dependson the head of water, i.e., the height of thewater in the reservoir above the level of thetube. The length of the tube can be readilyfound. The radius is more difficult to find.It has been measured for you by introducinga thread of mercury into the tube,determining its length and then its mass (andhence volume). The resulting values of theradii of the tubes are provided.

The viscosity of water changes rapidly with temperature (See tables in the references.). Measure thetemperature of the water at the beginning and end of the lab period. If there is a difference, estimatethe error this introduces into your determination of the viscosity.

CORRECTIONS

There are several empirical corrections that have been made to the Poiseuille formula. In particulara dynamical correction may be worth making in this experiment. It can be represented as follows:

p �8��

�a 4Q �

m�

�a 4Q 2

The first term on the right hand side is from the Poiseuille formula and the second term is thecorrection. The value of the parameter m for a given tube may be found by plotting p/Q versus Qto obtain a straight line, or by using the computer fit to a p versus Q plot. For your data, roughly howbig is the correction term relative to the first term in the equation? (Find numerical values for thetwo terms for typical Q values. What percentage of the first term is the correction term?)

UNITS

The C.G.S. unit of viscosity is the poise. The viscosity of water at room temperature is about onecentipoise. The S.I. unit could be expressed as N.s.m-2, but is not yet in common use.

(revised jbv - 89, tk -98)